Hybrid-type nodal ring phonons and coexistence of higher-order quadratic nodal line phonons in an AgZr alloy

Nodal line states in electronic systems can be classified based on the order of dispersion around a band degeneracy (i.e., linear, quadratic, or cubic) and the various slopes of crossing bands (i.e., type I, type II, critical type, and hybrid type). The concept of topology has been extended to phonons, promoting the birth of topological phonons. A series of nodal line phonons have been predicted in solid-state materials both theoretically and experimentally. However, the proposed nodal line phonons have a type-I or type-II linear phonon band dispersion. This study proves, based on first principles, that the hybrid-type linear nodal ring phonons and the higher-order quadratic nodal line (QNL) phonons coexist in one realistic solid-state material—AgZr with P4/nmm structure. Moreover, an effective Hamiltonian characterizing the hybrid nodal ring (HNR) phonons and the QNL phonons was established to explore the relevant physics. The AgZr material can be viewed as an ideal material to study the entanglement between HNR and QNL phonons.

Figure 1

(a) The crystal structures of AgZr with P4/nmm-type structures (space group No. 129). (b) The 3D coordinate system of BZ and some high-symmetry points.

Figure 2

(a) The phonon dispersion of AgZr along $\mathrm{\Gamma }-X-M-\mathrm{\Gamma }$–Z–R–A–Z–X–R–M–A paths. R1 and R2 are two regions of interest in this paper. In R2, a type-II phonon band crossing point P1 is visible. (b) Schematic figures of QNL along $\mathrm{\Gamma }-Z$ path and two Z-centered hybrid nodal rings (HNRs) in (110) and ($1\overline{1}0$) planes.

Figure 3

(a) Selected symmetry points in the $k_y=0$ plane. f1–f4 (d1–d4) are equally spaced between R and X (Z and $\mathrm{\Gamma }$). (b) Phonon dispersions along the $\mathrm{f}n–\mathrm{d}n–\mathrm{f}n$' ($n$=1, 2, 3, 4) paths; (c), (d) The 3D phononic bands in the $k_x$–$k_y=0$ plane and the color map in the $k_x$–$k_y=0$ plane. The color map shows the local gap between two crossing phonon bands. The QNL and the HNR phonons are denoted using black arrows.

Figure 4

(a) Schematic illustrations of HNR phonons and some selected $k$ paths in ($1\overline{1}0$) plane. The type-I and type-II nodal lines are denoted by red and blue lines, respectively. (b), (c) The calculated phonon dispersions along $Z–\mathrm{a}n$ and $Z–\mathrm{b}n$ ($n$=1, 2, 3, 4), respectively. Different types (types I and II) of phonon band dispersions around the phonon band crossing points (red circles) are also given.

Figure 5

(a) 3D bulk BZ and the surface BZ along the [010] direction of AgZr alloy. (c) The phonon dispersion of the AgZr along c–d–e path in 3D BZ. (b) and (d) Phonon surface spectra along $\overline{M}$–$\overline{Z}$–$\overline{A}$ and $\overline{c}$–$\overline{d}$–$\overline{e}$ path on [010] surface, respectively. The positions of phonon band-crossing points P1, P2, P3, and P4 are marked by white circles. P1 and P2 belong to HNR, and P3 and P4 belong to QNL.

Figure 6

Schematic figures of (a) type-I, (b) type-II, (c) critical-type, (d) linear, (e) quadratic, and (f) cubic band crossing points, respectively.